Fusion systems and group actions with abelian isotropy subgroups

Abstract

We prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × double struk S signn1 ×... × double struk S signnk for some positive integers n1, ..., nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology. Copyright © Edinburgh Mathematical Society 2013

Any finite group acts freely and homologically trivially on a product of spheres

“…The question as to whether any finite group acts freely and homologically trivially was mentioned as an open question in [13] and [14] and was mentioned by the author as an open problem in the problem session of the 2005 BIRS conference Homotopy and Group Actions. Motivated by this, Özgün Ünlü and Ergün Yalçın proved the following theorem.…”